Throughout the construction of the Mishkan (Tabernacle) and Bigdei Kodesh (holy clothing for the Kohanim), they use a mix of red, blue, and purple wool (plus [white?] linen, and in the case of the clothing, gold thread as well).

I'm sure there are several easy and obvious answers here, but for the sake of acknowledging the question (which I'm sure some kids must have asked):

4 Answers
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Even if the threads are fairly fine (and we don't know if they were), two colors plied together still looks like two colors, not the combined color. Thread is not like paint. Now even if at the usual viewing distance most people would see it as the combined color, it would not look that way close up, like to the kohein wearing the garment or tending to the curtains.

Also, as Gershon Gold noted, the red and blue might not be the right shades to produce the intended purple.

R' Hirsch (e.g. in the long comment at the end of Ex. 25:1-8) takes the four types of thread used in Mishkan construction to represent four basic aspects of life that we humans need to strive to perfect within ourselves and unify in the service of God:

Regardless of just how the juxtaposition of colors looked to the observer, an informed observer would, when encountering the holy objects made of this mixture, be sure to have the appropriate intentions regarding self-perfection in mind, and would use the symbolism present in the threads as a mnemonic that reinforces those intentions.

Waving a white, red, purple, and blue flag in front of a R' Hirsch fan is like waving a red flag in front of a bull (the symbol of devoting one's power to work hard to the service of God, by the way).
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Isaac MosesFeb 28 '12 at 2:26

The strings where mixed not just to get the desired color. Each detail in mishkan has its deep meaning about how G-d rules our world and so on. It's like asking: "Why we take 3 hadasim on Sukkos, one is already green enough?".